Inverse Stefan problem from indirect measurements, application to zircon crystallization

The growth of zircon crystals in cooling magmas is modelled by the one-phase Stefan problem, with the growth rate that depends on the magma cooling rate. Some rare elements (like e.g. U, Th, Hf) are incorporated in the crystal at trace concentration. These elements have different temperature-dependent diffusion and partition coefficients. As a consequence their final spatial repartition in the crystal depends on the temperature evolution of the magma during the cooling. The present work proposes to reconstruct the temperature evolution from the measurements of trace elements concentration in natural zircon. This inverse problem is solved by minimizing the misfit between calculated and measured trace element concentrations. The tangent model to the one-phase Stefan problem provides the sensitivity matrix, and the quadratic cost-function is minimized using Gauss–Newton method. The identifiability and the error on the retrieved parameters are studied in the framework of BLUE (Best Linear Unbiased Linear Estimator). The algorithm is tested on two synthetic datasets, and on real data obtained in a zircon crystal from early Fish Canyon tuff eruption. Reconstructed temperature ranges and cooling duration are in good agreement with available petrological interpretation.